Never More Than Two Degrees

The Hook

Two chapters ago the Rambam told us the strangest thing in this entire section of the Mishneh Torah. The sun moves at a perfectly uniform rate and always has, and we cannot see that rate, because we are not standing at the center of its orbit. What we see instead is a speed that runs fast and slow. He named the two of them, the mean motion and the true motion, and then left us holding a gap with no way to cross it.

Chapter thirteen is the bridge. And the astonishing thing about the bridge is how short it is. The entire discrepancy between where the sun would be if we stood at the center and where it actually appears in the sky never exceeds one degree and fifty nine minutes. Two degrees out of three hundred and sixty. At two positions in the circle the discrepancy is exactly nothing, and the mean position and the true position are the same number. He has spent chapters building an apparatus for a correction smaller than the width of four moons.

Mean Position, Apogee, Course

The procedure has three moves and he states them without a syllable of warm up. If you wish to know the true position of the sun on any day you desire, first calculate its mean position by the methods already explained. Then calculate the position of the apogee of the sun, the point of its orbit farthest from us. Then subtract the apogee from the mean position. What is left over is called the maslul, the course of the sun.

Sit with that for a moment, because it is the whole idea in one subtraction. The correction is not a function of where the sun is. It is a function of how far the sun has traveled from its farthest point. The apogee is the anchor. Everything the chapter does afterward is a measurement of distance from that anchor, and the number you carry forward is not a position at all. It is a displacement.

Then the second move, and it is the one people get wrong. From the course you derive an angle, and what you do with that angle depends on which half of the circle you are in. If the course is less than one hundred eighty degrees, the angle is subtracted from the mean position. If the course is more than one hundred eighty degrees, the angle is added. Same sun, same table, same arithmetic, and the correction pulls the sun backward through half of the year and pushes it forward through the other half.

And then the small ruling that most readers skate past. If the course is an even one hundred eighty degrees or an even three hundred sixty, there is no angle at all. The mean position is the true position. Which means the two moments when the abstraction and the observation agree perfectly are not moments of moderation. They are the two extremes, the farthest point and the nearest, the top and the bottom of the whole eccentric arrangement. Everywhere in between, the two descriptions disagree. It is only at the extremities that they coincide.

The Table, and the Sign That Flips

Then he simply hands you the table, seventeen rows of it, in the flattest voice imaginable. If the course is ten degrees the angle is twenty minutes. Twenty degrees, forty minutes. Thirty degrees, fifty eight minutes, and already the pattern has begun to bend, because two more degrees no longer buy two more minutes. Forty degrees, one degree and fifteen minutes. Sixty degrees, one degree and forty one. Ninety degrees, one degree and fifty nine, and that is the maximum the sun will ever ask of you. Then the curve turns over and comes back down. One hundred degrees, one degree and fifty eight. One hundred fifty, one degree and one minute. One hundred seventy, twenty one minutes. One hundred eighty, nothing.

What the table is describing is a quantity that grows quickly at first, slows as it climbs, stalls at its peak, and then unwinds. He does not name it or theorize about it. He tabulates it and moves on.

Past one hundred eighty degrees he does not extend the table at all. He tells you to fold the circle in half. Subtract the course from three hundred sixty and look up the remainder. A course of two hundred degrees becomes one hundred sixty, whose angle you already know is forty two minutes. A course of three hundred degrees becomes sixty, whose angle is one degree and forty one minutes. The second half of the circle is the first half read backward, with the sign reversed, and he expects you to see that and use it rather than memorize twice as much.

And for the degrees between the rows, he teaches interpolation with a worked example that takes four sentences. What if the course is sixty five degrees? You know that sixty gives one degree and forty one minutes and that seventy gives one degree and fifty one. There are ten minutes between the two measures across ten degrees, so a degree brings a minute, and sixty five gives one degree and forty six minutes. Sixty seven gives one degree and forty eight. Follow the same procedure, he adds, for any course with both units and tens, for the sun and for the moon alike.

What He Tells You to Throw Away

Now the part I keep coming back to. Having built a table calibrated to the minute, he immediately instructs you to degrade the number you feed into it.

He runs the whole method on a real date, the beginning of Friday night, the fourteenth of Tammuz of the year he was writing. The mean position of the sun is one hundred five degrees, thirty seven minutes, twenty five seconds. The apogee is eighty six degrees, forty five minutes, twenty three seconds. Subtract, and the course is eighteen degrees, fifty two minutes, two seconds. And then, mid sentence, the ruling. With regard to the course, the minutes are of no consequence. If they are less than thirty, disregard them entirely. If they are more than thirty, treat them as an additional degree. Fifty two minutes, so the course is nineteen degrees. Ten degrees gives twenty minutes and twenty degrees gives forty, two minutes per degree, so nineteen degrees gives thirty eight minutes.

The course was under one hundred eighty, so the thirty eight minutes come off the mean position, leaving one hundred four degrees, fifty nine minutes and twenty five seconds. Ninety degrees of that belong to the constellations already passed, so the sun stands in Cancer, fifteen degrees into it less thirty five seconds. And then the second instruction to discard. One need not pay attention to the seconds at all, not for the sun, not for the moon, not in any calculation regarding the sighting. If the seconds are around thirty or more, count them as a minute and add them in.

Read those two rulings together and something clarifies. He is not being sloppy, and he is not being humble about his instruments. He is specifying the precision of the operation. The seconds in the course cannot possibly matter, because a whole degree of course moves the angle by only a minute or two, and a minute of angle will never change whether a witness sees the moon on a given evening. Carrying those seconds forward would not make the answer better. It would only make the person doing the work slower, more anxious, and more likely to make a real mistake in the columns that actually decide the outcome. He said as much two chapters ago when he defended his approximations in advance: where the method is imprecise, the computation proved that the imprecision does not affect the determination of visibility.

He is doing something that takes more confidence than precision does. He is telling you exactly how much of your carefulness is real and how much of it is decoration.

And the chapter closes by quietly widening. Since you can now find the sun's location on any date you choose, you can also calculate the true date of any equinox or solstice you desire, whether one that will occur after the epoch he fixed or one that occurred in years long past. The method runs in both directions. The same handful of steps that tells the court when to look for the moon tells you when an autumn began before anyone alive was born.

The Unifying Principle

A chapter of arithmetic has made three claims that are not arithmetic at all.

The first is that the correction is small. One degree and fifty nine minutes at its very largest, and usually far less. The whole difference between the world as it truly runs and the world as it appears from where we are standing is about half a percent of a circle. Not a different sun. Not a different orbit. A slight and precisely calculable offset, produced entirely by the fact that we are not at the center.

The second is that the correction changes sign. The identical procedure that subtracts through one half of the circle adds through the other. Nothing about the sun reverses. What reverses is our relation to the point it is measured from. And the third is that the correction is zero at the extremes, so that the two positions where the ideal and the actual agree completely are the farthest and the nearest, and everywhere comfortable in between there is a gap.

The Alter Rebbe's line that a little light dispels a great deal of darkness is usually heard as encouragement, and here it becomes a description of a mechanism. The quantity is small and the effect is total. Without those thirty eight minutes the court's whole calculation lands on the wrong evening. There is no proportion between the size of a correction and the size of what it corrects.

The Baal Shem Tov taught that a soul may come down into a body and live seventy or eighty years in order to do a single favor for another person. That is the same arithmetic. An entire apparatus for a correction of two degrees. And the Rebbe returned again and again to the Rambam's own ruling in the laws of repentance, that a person should regard the world as exactly balanced between merit and liability, so that one deed tips the whole of it. Both teachings say what the table says. Do not read the size of the adjustment as a measure of its importance.

Modern Application

Most of what makes us wrong is not a gross error. It is an eccentricity, a small fixed offset built into where we are standing, applied consistently, and never corrected for because it never feels large enough to bother with. Two degrees. You would not notice two degrees in any single reading. Over a year of readings it puts you on the wrong day.

The Rambam's instruction is that this offset is knowable and tabulated and worth the trouble of applying, and that applying it requires you to first find your apogee, the point you are farthest from, and measure your distance from there. Not from where you feel yourself to be. From the extreme.

And there is the harder discipline in the second half of the chapter, which is knowing what to discard. We live in a culture with unlimited decimal places and almost no guidance about which ones matter. More data is assumed to be more truth. The Rambam refuses that. He proved, before he wrote, which of his own figures could not affect the outcome, and then he told his readers to drop them, and he gave a reason that has nothing to do with laziness: so that a person unpracticed in these matters will not become flustered by work that would not change the answer. Precision that does not change a decision is not rigor. It is a way of avoiding the decision.

The Closing

The Sfat Emet reads the moon as the model for Israel precisely because it is not a body that simply shines. It wanes, it needs correcting, it has to be watched for and testified about and calculated toward, and its renewal is an event rather than a constant. The sun in this chapter is the other half of that picture. It never varies at all and it still needs a correction of two degrees, not because anything is wrong with the sun, but because of where the observer happens to be.

Find the mean position, which is the truth. Find the apogee, which is the point you are farthest from. Subtract, and you have your course, which is not where you are but how far you have come from the extreme. Read the small correction off the table. Add it in one half of the circle and subtract it in the other, and at the two extremities apply nothing at all, because there the true and the mean are already the same. Then throw away the seconds, because they were never going to change the answer, and a person who cannot tell the difference between what matters and what merely can be measured will spend his life computing and still miss the moon.